MEASURING PROCESS QUALITY

ON AN ORDINAL SCALE BASIS

E. Bashkansky, T.GadrichIndustrial Engineering & Management Department

E.GodikSoftware Engineering Department

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PROCESS QUALITY

Gap Action

ProcessQuality Control

Target Measure

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Ordinal Variables in Quality Engineering

Quality sort Customer satisfaction grade Vendor’s priority Customer importance (QFD) Failure severity Internet page rank Vote result (pro, abstain, contra) the power of linkage (QFD) …

Traditional approach: assigning arbitrary numerical values to the different categories of the ordinal variable

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Quality variable having three levels of quality

Traditional Approach

Quality

level

Assigned value

H 9

M 3

L 1

Quality

level

Assigned value

H 3

M 2

L 1

“H” – High Quality “M” – Medium Quality “L” – Low Quality

H > M > L

Scale A Scale B

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Traditional Approach - Average

Sample HLL

According to A latent scale the average equals 1.67 positioning the average between Low and Medium quality

According to B latent scale the average equals 3.67 positioning the average between Medium and High quailty

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Study’s Purpose

Estimation the quality of a stable process without assigning any numerical values to the ordinal variables.

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MedianHHHHHHMMMLLLL

Advantage: Simple

Natural Measure for the Ordered Samples

Disadvantage: Robust

HHHMLLL MMMMLLL

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Quality measure of a given sample

Equals to the relative position of the given sample in a quality ladder that is built for a samples of the same size.

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The Rational of a Quality Ladder

Q -------- HH…H

--------

--------

-------- quality represented by a sample

--------

--------

--------

--------

--------

-------- LL…L

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Various possible quality ladders for a sample n=2

HH

HM

MM

HL

ML

LL

HH HM

HL

MM

ML LL

HH

HM

HL=MM

ML

LL

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F-function(Cumulative distribution function – CDF)

Define: Pi = proportion of products belonging to i - th quality level.

FL = PL ;

FM = PL + PM ;

FH = PL + PM + PH =1

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Graphical presentation of a different sample using F-space

R={LLLLL}

S={HHHHH}

T={MMMMM}

O={HHMLL}

P={HMMLL{

Q={HMMML}

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Ordinal dispersion

Blair, J., & Lacy, M. G. (2000)

2)5.0(

)1(,

iMLi

i FF

VAR

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Proposed Quality Ladders

1. Rank and dispersion (R&D)- based on Franceschini F. , Galetto M., Varetto M., Qual. Reliab. Engng. Int. 2005; 21:177–195

2. Median and Entourage (M&E)

3. Proportion Ratio and Dispersion (PR&D)

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1.Rank and dispersion criterion (R&D criterion)

The algorithm has two stages:

First stage sorts the samples in ascending order according to their ranks value .

Rank value = 0*(# L) + 1*(# M) + 2*(# H)

Second stage orders samples belonging to the same rank class according to their dispersion values in descending order. The ordered sample having larger dispersion is located at a lower position in the quality ladder.

Disadvantage

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1.Rank and dispersion criterion (R&D criterion) - example

No. SAMPLE Rank Variation1 HHH 6 02 HHM 5 0.53 HMM 4 0.54 HHL 4 15 MMM 3 06 HML 3 17 MML 2 0.58 HLL 2 19 MLL 1 0.510 LLL 0 0

Q

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Graphical interpretation of R&D criterion

FL

1

TS

R

O

450

Figure 4: Graphical illustration of rank and dispersion criterion

1

0.5

0.5

0

FM

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2.Median and Entourage Criterion (M&E criterion)-Example

n=3 n=2 n=1HHH HH HHHM HM MHHL MM LHMM HLMMM MLHML LLMMLHLLMLLLLL

Quality

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3.Proportion Ratio and Dispersion criterion (PR&D) - first stage

Define the proportion ratio (PR) as:

As the quality of the sample increases, the value of PR decreases, and vice versa.

So, first, samples are arranged according to their decreasing PR values.

PR

F

F

PP

PPPR

L

M

HM

ML 01

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Graphical illustration of PR criterion

FL

FM

Figure 6: Graphical illustration for Proportion ratio criterion

TS

R

1

Z

0

1Q

09001

tgF

FPR

QL

QM

Q

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Comparison between various criterions for a sample size n=3

No. R&D M&E PR&D1 HHH HHH HHH2 HHM HHM HHM3 HMM HHL HHL4 HHL HMM HMM5 MMM MMM MMM6 HML HML HML7 MML MML MML8 HLL HLL HLL9 MLL MLL MLL10 LLL LLL LLL

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Verification of proposed criterions

sample (n=100) vs. infinite population

7.01.02.0 HMLPPP

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Verification of proposed criterions

sample (n=100) vs. infinite population

5.02.03.0 HMLPPP

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Verification of proposed criterionsrelative position of the sample (n=100) mode quality vs. relative quality position in the infinite population

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Verification of proposed criterionsrelative position of the sample ( n=10 ) mode quality vs.

relative quality position in the finite ( N=100 ) population

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Verification of proposed measures:relative position of the sample ( n=10 ) median quality vs. relative quality position in the finite ( N=100 ) population

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Thank You